Pure Interchange Modes and Full MHD Normal Mods

The ideal-MHD energy principle can be used to estimate the growth rate of an interchange instability, based on the change in potential energy associated with an exchange of adjacent flux tubes, assuming pressure balance along the field line. In a system that is interchange stable, the same procedure can be used to estimate the frequency of an interchange oscillation, along with its eigenfunctions. Using this kind of procedure, we have computed pure-interchange eigenfrequencies and eigenfunctions for a thin ideal-MHD thin filament moving in the midnight plane of an average magnetosphere. Of course, in a real system undergoing an interchange-type oscillation, the pressure is only constant along the field line if the frequency of the interchange oscillation is much less than the frequency of the slow mode that tries to maintain pressure equilibrium. We also have computed low-frequency full-MHD thin-filament normal modes for the same average magnetosphere. In every case with L>2, one full-MHD normal mode can be identified as being closest to an ideal-MHD interchange mode. For field lines that extend more than 10 R_E in the tail, the eigenfrequencies and eigenfunctions for the ideal-interchange modes and full-MHD modes are in good agreement, but they become different in the inner magnetosphere, particularly in the plasmasphere, where the full-MHD modes have noticeably lower frequencies, and pressure is not close to being constant along field lines. However, the use of ideal single-fluid MHD is questionable in the plasmasphere, where the mass density is mainly in the cold component, and the pressure is primarily in the ring current.